[Math] How to find the length of the third side of any triangle

trianglestrigonometry

I will know the length of two sides of any triangle that I use, but I will not know any of the angles. I know how to find the length of the third side if I knew the angle where I am sitting, but how can I quickly find the included angle where I am sitting with basic geometric tools or something else?

Best Answer

Basic Answer:

You can't.

Here's what you can know

Given only the length of two sides of a triangle, the length of the third side is not fixed. Let a and b represent the lengths of the two known sides such that $a \geq b$. Let c represent the length of the unknown side, the length of c must fall within

$a - b < c < a + b$

Based on the example triangle you give, the third side, c, must be

\begin{align} 475 - 390 <& c < 475 + 390 \\ 85 <& c < 865 \end{align}

More Detail

With the two given lengths, we can construct a segment and a circle. It is irrelevant where we choose to position the segment. It's endpoints can lie anywhere with the caveat that it has a length exactly equal to the first known side. For your example let the segment have length 475.

Segment

With the second side we can represent all possible endpoints as the points of a circle. This makes sense since a circle is the set of all points a given distance away from the center point. So the circle would have radius 390.

Segment + Circle

Well, you want to know the third side of the triangle, but the third side -- without any other information about the triangle -- could be any segment which starts at the free endpoint of our original segment and has its other endpoint on the circle. You can see why this means that there is more than one possible segment length because not all such segments have equal length. Here are a few examples.

ex. 1 ex. 2 ex. 3

Closing Remarks

If you want to calculate the third side of the triangle, you need more information than simply two sides. For example, if you know the triangle is a right triangle, or if you know the measure of the included angle between the two known segments, then you can determine the length of the third side.

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